    May  2018, 17(3): 787-806. doi: 10.3934/cpaa.2018040

## Symmetry and asymptotic behavior of ground state solutions for schrödinger systems with linear interaction

 HLM, CEMS, Academy of Mathematics and Systems Science, the Chinese, Academy of Sciences, Beijing 100190, School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author

Received  June 2017 Revised  June 2017 Published  January 2018

Fund Project: Supported by the National Natural Science Foundation of China 11325107,11331010,11771428.

We study symmetry and asymptotic behavior of ground state solutions for the doubly coupled nonlinear Schrödinger elliptic system
 $\left\{ {\begin{array}{*{20}{l}}{ - \Delta u + {\lambda _1}u + \kappa v = {\mu _1}{u^3} + \beta u{v^2}}&{\quad {\rm{ in}}\;\;\Omega ,}\\{ - \Delta v + {\lambda _2}v + \kappa u = {\mu _2}{v^3} + \beta {u^2}v}&{\quad {\rm{ in}}\;\;\Omega ,}\\{u = v = 0\;on\;\;\partial \Omega \;({\rm{or}}\;u,v \in {H^1}({\mathbb{R}^N})\;{\rm{as}}\;\Omega = {\mathbb{R}^N}),}&{}\end{array}} \right.$
where
 $N≤3, Ω\subseteq\mathbb{R}^N$
is a smooth domain. First we establish the symmetry of ground state solutions, that is, when
 $Ω$
is radially symmetric, we show that ground state solution is foliated Schwarz symmetric with respect to the same point. Moreover, ground state solutions must be radially symmetric under the condition that
 $Ω$
is a ball or the whole space
 $\mathbb{R}^N$
. Next we investigate the asymptotic behavior of positive ground state solution as
 $κ\to 0^-$
, which shows that the limiting profile is exactly a minimizer for
 $c_0$
(the minimized energy constrained on Nehari manifold corresponds to the limit systems). Finally for a system with three equations, we prove the existence of ground state solutions whose all components are nonzero.
Citation: Zhitao Zhang, Haijun Luo. Symmetry and asymptotic behavior of ground state solutions for schrödinger systems with linear interaction. Communications on Pure & Applied Analysis, 2018, 17 (3) : 787-806. doi: 10.3934/cpaa.2018040
##### References:
  N. Akhmediev and A. Ankiewicz, Partially coherent soltions on a finite background, Phys. Rev. Lett., 82 (1999), 2661-2664.  doi: 10.1103/PhysRevLett.82.2661. Google Scholar  A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458. Google Scholar  A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82. Google Scholar  T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differential Equations, 19 (2006), 200-207. Google Scholar  T. Bartsch, Z. Q. Wang and J. C. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367. Google Scholar  J. Belmonte-Beitia, V. M. Pérez-García and P. J. Torres, Solitary waves for linearly coupled nonlinear Schrödinger equations with inhomogeneous coefficients, J. Nonlinear Sci., 19 (2009), 437-451. Google Scholar  H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. Google Scholar  J. E. Brothers and W. P. Ziemer, Minimal rearrangements of Sobolev functions, Acta Univ. Carolin. Math. Phys., 28 (1987), 13-24. Google Scholar  J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56. Google Scholar  B. Deconinck, Linearly coupled Bose-Einstein condesates: From Rabi oscillations and quasiperiodic solutions to oscillating domain walls and spiral waves, Phys. Rev. A, 70 (2004), 705-706.  doi: 10.1103/PhysRevA.70.063605. Google Scholar  G. W. Dai, R. S. Tian and Z. T. Zhang, Global bifurcation, priori bounds and uniqueness of positive solutions for coupled nonlinear Schrödinger systems, preprint. Google Scholar  D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 ed., Springer-Verlag, Berlin, 2001. Google Scholar  D. S. Hall, M. R. Matthews, J. R. Ensher and C. E. Wieman, Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 1539-1542.  doi: 10.1103/PhysRevLett.81.1539. Google Scholar  N. Ikoma and K. Tanaka, A local mountain pass type result for a system of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 40 (2011), 449-480. Google Scholar  K. Li and Z. T. Zhang, Existence of solutions for a Schrödinger system with linear and nonlinear couplings, J. Math. Phys., 57 (2016), 17 pp. Google Scholar  T. C. Lin and J. C. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^n, n≤3$, Comm. Math. Phys., 255 (2005), 629-653. Google Scholar  Z. L. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731. Google Scholar  L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767. Google Scholar  C. J. Myatt, Production of two overlapping Bose-Einstein condensates by sympathetic cooling, Phys. Rev. Lett., 78 (1997), 586-589.  doi: 10.1103/PhysRevLett.78.586. Google Scholar  B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302. Google Scholar  Ch. Rüegg, Bose-Einstein condensate of the triplet ststes in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.  doi: 10.1038/nature01617. Google Scholar  B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^n$, Comm. Math. Phys., 271 (2007), 199-221. Google Scholar  H. Tavares and T. Weth, Existence and symmetry results for competing variational systems, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 715-740. Google Scholar  R. S. Tian and Z. T. Zhang, Existence and bifurcation of solutions for a double coupled system of Schrödinger equations, Sci. China Math., 58 (2015), 1607-1620. Google Scholar  E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.  doi: 10.1103/PhysRevLett.81.5718. Google Scholar  W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations, 42 (1981), 400-413. Google Scholar  Z. Q. Wang and M. Willem, Partial symmetry of vector solutions for elliptic systems, J. Anal. Math., 122 (2014), 69-85. Google Scholar  J. C. Wei and T. Weth, Nonradial symmetric bound states for a system of coupled Schrödinger equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 18 (2007), 279-293. Google Scholar  J. C. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011. Google Scholar  T. Weth, Symmetry of solutions to variational problems for nonlinear elliptic equations via reflection methods, Jahresber. Dtsch. Math.-Ver., 112 (2010), 119-158. Google Scholar  M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. Google Scholar  M. Willem, Principles d'analyse fonctionnelle, Cassini, Pairs, 2007. Google Scholar

show all references

##### References:
  N. Akhmediev and A. Ankiewicz, Partially coherent soltions on a finite background, Phys. Rev. Lett., 82 (1999), 2661-2664.  doi: 10.1103/PhysRevLett.82.2661. Google Scholar  A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458. Google Scholar  A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82. Google Scholar  T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differential Equations, 19 (2006), 200-207. Google Scholar  T. Bartsch, Z. Q. Wang and J. C. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367. Google Scholar  J. Belmonte-Beitia, V. M. Pérez-García and P. J. Torres, Solitary waves for linearly coupled nonlinear Schrödinger equations with inhomogeneous coefficients, J. Nonlinear Sci., 19 (2009), 437-451. Google Scholar  H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. Google Scholar  J. E. Brothers and W. P. Ziemer, Minimal rearrangements of Sobolev functions, Acta Univ. Carolin. Math. Phys., 28 (1987), 13-24. Google Scholar  J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56. Google Scholar  B. Deconinck, Linearly coupled Bose-Einstein condesates: From Rabi oscillations and quasiperiodic solutions to oscillating domain walls and spiral waves, Phys. Rev. A, 70 (2004), 705-706.  doi: 10.1103/PhysRevA.70.063605. Google Scholar  G. W. Dai, R. S. Tian and Z. T. Zhang, Global bifurcation, priori bounds and uniqueness of positive solutions for coupled nonlinear Schrödinger systems, preprint. Google Scholar  D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 ed., Springer-Verlag, Berlin, 2001. Google Scholar  D. S. Hall, M. R. Matthews, J. R. Ensher and C. E. Wieman, Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 1539-1542.  doi: 10.1103/PhysRevLett.81.1539. Google Scholar  N. Ikoma and K. Tanaka, A local mountain pass type result for a system of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 40 (2011), 449-480. Google Scholar  K. Li and Z. T. Zhang, Existence of solutions for a Schrödinger system with linear and nonlinear couplings, J. Math. Phys., 57 (2016), 17 pp. Google Scholar  T. C. Lin and J. C. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbb{R}^n, n≤3$, Comm. Math. Phys., 255 (2005), 629-653. Google Scholar  Z. L. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731. Google Scholar  L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767. Google Scholar  C. J. Myatt, Production of two overlapping Bose-Einstein condensates by sympathetic cooling, Phys. Rev. Lett., 78 (1997), 586-589.  doi: 10.1103/PhysRevLett.78.586. Google Scholar  B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure Appl. Math., 63 (2010), 267-302. Google Scholar  Ch. Rüegg, Bose-Einstein condensate of the triplet ststes in the magnetic insulator TlCuCl3, Nature, 423 (2003), 62-65.  doi: 10.1038/nature01617. Google Scholar  B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^n$, Comm. Math. Phys., 271 (2007), 199-221. Google Scholar  H. Tavares and T. Weth, Existence and symmetry results for competing variational systems, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 715-740. Google Scholar  R. S. Tian and Z. T. Zhang, Existence and bifurcation of solutions for a double coupled system of Schrödinger equations, Sci. China Math., 58 (2015), 1607-1620. Google Scholar  E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.  doi: 10.1103/PhysRevLett.81.5718. Google Scholar  W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations, 42 (1981), 400-413. Google Scholar  Z. Q. Wang and M. Willem, Partial symmetry of vector solutions for elliptic systems, J. Anal. Math., 122 (2014), 69-85. Google Scholar  J. C. Wei and T. Weth, Nonradial symmetric bound states for a system of coupled Schrödinger equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 18 (2007), 279-293. Google Scholar  J. C. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011. Google Scholar  T. Weth, Symmetry of solutions to variational problems for nonlinear elliptic equations via reflection methods, Jahresber. Dtsch. Math.-Ver., 112 (2010), 119-158. Google Scholar  M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. Google Scholar  M. Willem, Principles d'analyse fonctionnelle, Cassini, Pairs, 2007. Google Scholar
  Jian Zhang, Wen Zhang, Xianhua Tang. Ground state solutions for Hamiltonian elliptic system with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4565-4583. doi: 10.3934/dcds.2017195  Jian Zhang, Wen Zhang. Existence and decay property of ground state solutions for Hamiltonian elliptic system. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2433-2455. doi: 10.3934/cpaa.2019110  Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329  Francisco Ortegón Gallego, María Teresa González Montesinos. Existence of a capacity solution to a coupled nonlinear parabolic--elliptic system. Communications on Pure & Applied Analysis, 2007, 6 (1) : 23-42. doi: 10.3934/cpaa.2007.6.23  Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179  Norihisa Ikoma. Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 943-966. doi: 10.3934/dcds.2015.35.943  Dengfeng Lü. Existence and concentration behavior of ground state solutions for magnetic nonlinear Choquard equations. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1781-1795. doi: 10.3934/cpaa.2016014  Zhanping Liang, Yuanmin Song, Fuyi Li. Positive ground state solutions of a quadratically coupled schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048  Xiaoping Wang. Ground state homoclinic solutions for a second-order Hamiltonian system. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2163-2175. doi: 10.3934/dcdss.2019139  Manuel del Pino, Michal Kowalczyk, Juncheng Wei. The Jacobi-Toda system and foliated interfaces. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 975-1006. doi: 10.3934/dcds.2010.28.975  Chun Shen, Wancheng Sheng, Meina Sun. The asymptotic limits of solutions to the Riemann problem for the scaled Leroux system. Communications on Pure & Applied Analysis, 2018, 17 (2) : 391-411. doi: 10.3934/cpaa.2018022  Haiyang He. Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2393-2408. doi: 10.3934/cpaa.2013.12.2393  Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074  Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991  Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations & Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023  Chuangye Liu, Zhi-Qiang Wang. A complete classification of ground-states for a coupled nonlinear Schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 115-130. doi: 10.3934/cpaa.2017005  Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193  Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure & Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015  Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091  Chao Ji. Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6071-6089. doi: 10.3934/dcdsb.2019131

2018 Impact Factor: 0.925